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  <section id="module-sympy.geometry.plane">
<span id="plane"></span><h1>Plane<a class="headerlink" href="#module-sympy.geometry.plane" title="Permalink to this headline">¶</a></h1>
<dl class="py class">
<dt class="sig sig-object py" id="sympy.geometry.plane.Plane">
<em class="property"><span class="pre">class</span> </em><span class="sig-prename descclassname"><span class="pre">sympy.geometry.plane.</span></span><span class="sig-name descname"><span class="pre">Plane</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">p1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/geometry/plane.py#L25-L886"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.geometry.plane.Plane" title="Permalink to this definition">¶</a></dt>
<dd><p>A plane is a flat, two-dimensional surface. A plane is the two-dimensional
analogue of a point (zero-dimensions), a line (one-dimension) and a solid
(three-dimensions). A plane can generally be constructed by two types of
inputs. They are three non-collinear points and a point and the plane’s
normal vector.</p>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Plane</span><span class="p">,</span> <span class="n">Point3D</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Point3D</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">Point3D</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="go">Plane(Point3D(1, 1, 1), (-1, 2, -1))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">Plane</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="go">Plane(Point3D(1, 1, 1), (-1, 2, -1))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">7</span><span class="p">))</span>
<span class="go">Plane(Point3D(1, 1, 1), (1, 4, 7))</span>
</pre></div>
</div>
<p class="rubric">Attributes</p>
<table class="docutils align-default">
<colgroup>
<col style="width: 57%" />
<col style="width: 43%" />
</colgroup>
<tbody>
<tr class="row-odd"><td><p>p1</p></td>
<td></td>
</tr>
<tr class="row-even"><td><p>normal_vector</p></td>
<td></td>
</tr>
</tbody>
</table>
<dl class="py method">
<dt class="sig sig-object py" id="sympy.geometry.plane.Plane.angle_between">
<span class="sig-name descname"><span class="pre">angle_between</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">o</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/geometry/plane.py#L89-L134"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.geometry.plane.Plane.angle_between" title="Permalink to this definition">¶</a></dt>
<dd><p>Angle between the plane and other geometric entity.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><p><strong>LinearEntity3D, Plane.</strong></p>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p><strong>angle</strong> : angle in radians</p>
</dd>
</dl>
<p class="rubric">Notes</p>
<p>This method accepts only 3D entities as it’s parameter, but if you want
to calculate the angle between a 2D entity and a plane you should
first convert to a 3D entity by projecting onto a desired plane and
then proceed to calculate the angle.</p>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point3D</span><span class="p">,</span> <span class="n">Line3D</span><span class="p">,</span> <span class="n">Plane</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Line3D</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">Point3D</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">angle_between</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
<span class="go">-asin(sqrt(21)/6)</span>
</pre></div>
</div>
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="sympy.geometry.plane.Plane.arbitrary_point">
<span class="sig-name descname"><span class="pre">arbitrary_point</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">u</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">v</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/geometry/plane.py#L137-L192"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.geometry.plane.Plane.arbitrary_point" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns an arbitrary point on the Plane. If given two
parameters, the point ranges over the entire plane. If given 1
or no parameters, returns a point with one parameter which,
when varying from 0 to 2*pi, moves the point in a circle of
radius 1 about p1 of the Plane.</p>
<dl class="field-list simple">
<dt class="field-odd">Returns</dt>
<dd class="field-odd"><p>Point3D</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Plane</span><span class="p">,</span> <span class="n">Ray</span>
<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">r</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="go">Point3D(1, u + 1, v + 1)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<span class="go">Point3D(1, cos(t) + 1, sin(t) + 1)</span>
</pre></div>
</div>
<p>While arbitrary values of u and v can move the point anywhere in
the plane, the single-parameter point can be used to construct a
ray whose arbitrary point can be located at angle t and radius
r from p.p1:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">Ray</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">p1</span><span class="p">,</span> <span class="n">_</span><span class="p">)</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
<span class="go">Point3D(1, r*cos(t) + 1, r*sin(t) + 1)</span>
</pre></div>
</div>
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="sympy.geometry.plane.Plane.are_concurrent">
<em class="property"><span class="pre">static</span> </em><span class="sig-name descname"><span class="pre">are_concurrent</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">planes</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/geometry/plane.py#L195-L242"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.geometry.plane.Plane.are_concurrent" title="Permalink to this definition">¶</a></dt>
<dd><p>Is a sequence of Planes concurrent?</p>
<p>Two or more Planes are concurrent if their intersections
are a common line.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><p><strong>planes: list</strong></p>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p>Boolean</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Plane</span><span class="p">,</span> <span class="n">Point3D</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">9</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">Plane</span><span class="o">.</span><span class="n">are_concurrent</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">Plane</span><span class="o">.</span><span class="n">are_concurrent</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
<span class="go">False</span>
</pre></div>
</div>
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="sympy.geometry.plane.Plane.distance">
<span class="sig-name descname"><span class="pre">distance</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">o</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/geometry/plane.py#L245-L297"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.geometry.plane.Plane.distance" title="Permalink to this definition">¶</a></dt>
<dd><p>Distance between the plane and another geometric entity.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><p><strong>Point3D, LinearEntity3D, Plane.</strong></p>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p>distance</p>
</dd>
</dl>
<p class="rubric">Notes</p>
<p>This method accepts only 3D entities as it’s parameter, but if you want
to calculate the distance between a 2D entity and a plane you should
first convert to a 3D entity by projecting onto a desired plane and
then proceed to calculate the distance.</p>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point3D</span><span class="p">,</span> <span class="n">Line3D</span><span class="p">,</span> <span class="n">Plane</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
<span class="go">sqrt(3)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">Line3D</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
<span class="go">0</span>
</pre></div>
</div>
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="sympy.geometry.plane.Plane.equals">
<span class="sig-name descname"><span class="pre">equals</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">o</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/geometry/plane.py#L300-L323"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.geometry.plane.Plane.equals" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns True if self and o are the same mathematical entities.</p>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Plane</span><span class="p">,</span> <span class="n">Point3D</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
<span class="go">False</span>
</pre></div>
</div>
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="sympy.geometry.plane.Plane.equation">
<span class="sig-name descname"><span class="pre">equation</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">z</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/geometry/plane.py#L326-L345"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.geometry.plane.Plane.equation" title="Permalink to this definition">¶</a></dt>
<dd><p>The equation of the Plane.</p>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point3D</span><span class="p">,</span> <span class="n">Plane</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">Point3D</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">7</span><span class="p">),</span> <span class="n">Point3D</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">equation</span><span class="p">()</span>
<span class="go">-23*x + 11*y - 2*z + 16</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">equation</span><span class="p">()</span>
<span class="go">6*x + 6*y + 6*z - 42</span>
</pre></div>
</div>
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="sympy.geometry.plane.Plane.intersection">
<span class="sig-name descname"><span class="pre">intersection</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">o</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/geometry/plane.py#L348-L431"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.geometry.plane.Plane.intersection" title="Permalink to this definition">¶</a></dt>
<dd><p>The intersection with other geometrical entity.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><p><strong>Point, Point3D, LinearEntity, LinearEntity3D, Plane</strong></p>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p>List</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point3D</span><span class="p">,</span> <span class="n">Line3D</span><span class="p">,</span> <span class="n">Plane</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
<span class="go">[Point3D(1, 2, 3)]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">Line3D</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">7</span><span class="p">),</span> <span class="n">Point3D</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
<span class="go">[Point3D(2, 2, 2)]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
<span class="go">[Line3D(Point3D(78/23, -24/23, 0), Point3D(147/23, 321/23, 23))]</span>
</pre></div>
</div>
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="sympy.geometry.plane.Plane.is_coplanar">
<span class="sig-name descname"><span class="pre">is_coplanar</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">o</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/geometry/plane.py#L434-L457"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.geometry.plane.Plane.is_coplanar" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns True if <span class="math notranslate nohighlight">\(o\)</span> is coplanar with self, else False.</p>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Plane</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">o</span> <span class="o">=</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">p2</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">==</span> <span class="n">p2</span>
<span class="go">False</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">is_coplanar</span><span class="p">(</span><span class="n">p2</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="sympy.geometry.plane.Plane.is_parallel">
<span class="sig-name descname"><span class="pre">is_parallel</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">l</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/geometry/plane.py#L460-L498"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.geometry.plane.Plane.is_parallel" title="Permalink to this definition">¶</a></dt>
<dd><p>Is the given geometric entity parallel to the plane?</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><p><strong>LinearEntity3D or Plane</strong></p>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p>Boolean</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Plane</span><span class="p">,</span> <span class="n">Point3D</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">6</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">12</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">is_parallel</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="sympy.geometry.plane.Plane.is_perpendicular">
<span class="sig-name descname"><span class="pre">is_perpendicular</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">l</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/geometry/plane.py#L501-L540"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.geometry.plane.Plane.is_perpendicular" title="Permalink to this definition">¶</a></dt>
<dd><p>is the given geometric entity perpendicualar to the given plane?</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><p><strong>LinearEntity3D or Plane</strong></p>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p>Boolean</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Plane</span><span class="p">,</span> <span class="n">Point3D</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">6</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">is_perpendicular</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>

<dl class="py property">
<dt class="sig sig-object py" id="sympy.geometry.plane.Plane.normal_vector">
<em class="property"><span class="pre">property</span> </em><span class="sig-name descname"><span class="pre">normal_vector</span></span><a class="headerlink" href="#sympy.geometry.plane.Plane.normal_vector" title="Permalink to this definition">¶</a></dt>
<dd><p>Normal vector of the given plane.</p>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point3D</span><span class="p">,</span> <span class="n">Plane</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Point3D</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">Point3D</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">normal_vector</span>
<span class="go">(-1, 2, -1)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">7</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">normal_vector</span>
<span class="go">(1, 4, 7)</span>
</pre></div>
</div>
</dd></dl>

<dl class="py property">
<dt class="sig sig-object py" id="sympy.geometry.plane.Plane.p1">
<em class="property"><span class="pre">property</span> </em><span class="sig-name descname"><span class="pre">p1</span></span><a class="headerlink" href="#sympy.geometry.plane.Plane.p1" title="Permalink to this definition">¶</a></dt>
<dd><p>The only defining point of the plane. Others can be obtained from the
arbitrary_point method.</p>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point3D</span><span class="p">,</span> <span class="n">Plane</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Point3D</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">Point3D</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">p1</span>
<span class="go">Point3D(1, 1, 1)</span>
</pre></div>
</div>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p><a class="reference internal" href="points.html#sympy.geometry.point.Point3D" title="sympy.geometry.point.Point3D"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sympy.geometry.point.Point3D</span></code></a></p>
</div>
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="sympy.geometry.plane.Plane.parallel_plane">
<span class="sig-name descname"><span class="pre">parallel_plane</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">pt</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/geometry/plane.py#L581-L605"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.geometry.plane.Plane.parallel_plane" title="Permalink to this definition">¶</a></dt>
<dd><p>Plane parallel to the given plane and passing through the point pt.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><p><strong>pt: Point3D</strong></p>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p>Plane</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Plane</span><span class="p">,</span> <span class="n">Point3D</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">parallel_plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
<span class="go">Plane(Point3D(2, 3, 5), (2, 4, 6))</span>
</pre></div>
</div>
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="sympy.geometry.plane.Plane.parameter_value">
<span class="sig-name descname"><span class="pre">parameter_value</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">other</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">u</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">v</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/geometry/plane.py#L824-L882"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.geometry.plane.Plane.parameter_value" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the parameter(s) corresponding to the given point.</p>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span>
<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Plane</span>
<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">t</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
</pre></div>
</div>
<p>By default, the parameter value returned defines a point
that is a distance of 1 from the Plane’s p1 value and
in line with the given point:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">on_circle</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">on_circle</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">p1</span><span class="p">)</span>
<span class="go">1</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">parameter_value</span><span class="p">(</span><span class="n">on_circle</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
<span class="go">{t: pi/4}</span>
</pre></div>
</div>
<p>Moving the point twice as far from p1 does not change
the parameter value:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">off_circle</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">p1</span> <span class="o">+</span> <span class="p">(</span><span class="n">on_circle</span> <span class="o">-</span> <span class="n">p</span><span class="o">.</span><span class="n">p1</span><span class="p">)</span><span class="o">*</span><span class="mi">2</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">off_circle</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">p1</span><span class="p">)</span>
<span class="go">2</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">parameter_value</span><span class="p">(</span><span class="n">off_circle</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
<span class="go">{t: pi/4}</span>
</pre></div>
</div>
<p>If the 2-value parameter is desired, supply the two
parameter symbols and a replacement dictionary will
be returned:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">parameter_value</span><span class="p">(</span><span class="n">on_circle</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="go">{u: sqrt(10)/10, v: sqrt(10)/30}</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">parameter_value</span><span class="p">(</span><span class="n">off_circle</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="go">{u: sqrt(10)/5, v: sqrt(10)/15}</span>
</pre></div>
</div>
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="sympy.geometry.plane.Plane.perpendicular_line">
<span class="sig-name descname"><span class="pre">perpendicular_line</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">pt</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/geometry/plane.py#L607-L630"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.geometry.plane.Plane.perpendicular_line" title="Permalink to this definition">¶</a></dt>
<dd><p>A line perpendicular to the given plane.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><p><strong>pt: Point3D</strong></p>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p>Line3D</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Plane</span><span class="p">,</span> <span class="n">Point3D</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">6</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">perpendicular_line</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">7</span><span class="p">))</span>
<span class="go">Line3D(Point3D(9, 8, 7), Point3D(11, 12, 13))</span>
</pre></div>
</div>
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="sympy.geometry.plane.Plane.perpendicular_plane">
<span class="sig-name descname"><span class="pre">perpendicular_plane</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">pts</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/geometry/plane.py#L632-L696"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.geometry.plane.Plane.perpendicular_plane" title="Permalink to this definition">¶</a></dt>
<dd><p>Return a perpendicular passing through the given points. If the
direction ratio between the points is the same as the Plane’s normal
vector then, to select from the infinite number of possible planes,
a third point will be chosen on the z-axis (or the y-axis
if the normal vector is already parallel to the z-axis). If less than
two points are given they will be supplied as follows: if no point is
given then pt1 will be self.p1; if a second point is not given it will
be a point through pt1 on a line parallel to the z-axis (if the normal
is not already the z-axis, otherwise on the line parallel to the
y-axis).</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><p><strong>pts: 0, 1 or 2 Point3D</strong></p>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p>Plane</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Plane</span><span class="p">,</span> <span class="n">Point3D</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">Point3D</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point3D</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">Z</span> <span class="o">=</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">normal_vector</span><span class="o">=</span><span class="n">Z</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">perpendicular_plane</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="go">Plane(Point3D(0, 0, 0), (1, 0, 0))</span>
</pre></div>
</div>
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="sympy.geometry.plane.Plane.projection">
<span class="sig-name descname"><span class="pre">projection</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">pt</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/geometry/plane.py#L750-L787"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.geometry.plane.Plane.projection" title="Permalink to this definition">¶</a></dt>
<dd><p>Project the given point onto the plane along the plane normal.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><p><strong>Point or Point3D</strong></p>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p>Point3D</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Plane</span><span class="p">,</span> <span class="n">Point3D</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
</pre></div>
</div>
<p>The projection is along the normal vector direction, not the z
axis, so (1, 1) does not project to (1, 1, 2) on the plane A:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">projection</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
<span class="go">Point3D(5/3, 5/3, 2/3)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span> <span class="ow">in</span> <span class="n">A</span>
<span class="go">True</span>
</pre></div>
</div>
<p>But the point (1, 1, 2) projects to (1, 1) on the XY-plane:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">XY</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">XY</span><span class="o">.</span><span class="n">projection</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="go">Point3D(1, 1, 0)</span>
</pre></div>
</div>
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="sympy.geometry.plane.Plane.projection_line">
<span class="sig-name descname"><span class="pre">projection_line</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">line</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/geometry/plane.py#L698-L748"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.geometry.plane.Plane.projection_line" title="Permalink to this definition">¶</a></dt>
<dd><p>Project the given line onto the plane through the normal plane
containing the line.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><p><strong>LinearEntity or LinearEntity3D</strong></p>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p>Point3D, Line3D, Ray3D or Segment3D</p>
</dd>
</dl>
<p class="rubric">Notes</p>
<p>For the interaction between 2D and 3D lines(segments, rays), you should
convert the line to 3D by using this method. For example for finding the
intersection between a 2D and a 3D line, convert the 2D line to a 3D line
by projecting it on a required plane and then proceed to find the
intersection between those lines.</p>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Plane</span><span class="p">,</span> <span class="n">Line</span><span class="p">,</span> <span class="n">Line3D</span><span class="p">,</span> <span class="n">Point3D</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Point3D</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">projection_line</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
<span class="go">Line3D(Point3D(4/3, 4/3, 1/3), Point3D(5/3, 5/3, -1/3))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">Line3D</span><span class="p">(</span><span class="n">Point3D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Point3D</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">projection_line</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
<span class="go">Point3D(1, 1, 1)</span>
</pre></div>
</div>
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="sympy.geometry.plane.Plane.random_point">
<span class="sig-name descname"><span class="pre">random_point</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">seed</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/geometry/plane.py#L789-L822"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.geometry.plane.Plane.random_point" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns a random point on the Plane.</p>
<dl class="field-list simple">
<dt class="field-odd">Returns</dt>
<dd class="field-odd"><p>Point3D</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Plane</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Plane</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">normal_vector</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">random_point</span><span class="p">(</span><span class="n">seed</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>  <span class="c1"># seed value is optional</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="go">Point3D(2.29, 0, -1.35)</span>
</pre></div>
</div>
<p>The random point can be moved to lie on the circle of radius
1 centered on p1:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">p1</span> <span class="o">+</span> <span class="p">(</span><span class="n">r</span> <span class="o">-</span> <span class="n">p</span><span class="o">.</span><span class="n">p1</span><span class="p">)</span><span class="o">.</span><span class="n">unit</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">p1</span><span class="p">)</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>

</dd></dl>

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